Topics. 2D Signal. a b. c d. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2004 Lecture 4 2D Fourier Transforms
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1 Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2004 Lecture 4 2D Fourier Transforms Topics. 2D Signal Representations 2. 2D Fourier Transform 3. Transform Pairs 4. FT Properties 2D Signal a b a b + = + c d + c d
2 Image Decomposition 0 a b 0 a b + = + c d c d + g[m,n] = aδ[m,n] + bδ[m,n ] + cδ[m,n] + dδ[m,n ] = g[k,l] δ[m k,n l] k= 0 l= 0 k= 0 l= 0 = c k,l b k,l [m,n] Orthonormal Basis Functions Discrete b k,l,b k, l = b k,l [m,n] b k, l [m,n] Continuous b kx,b k x, m= n= = δ[k k,l l ] k y = b kx = δ(k x k x k y (x, yb k x, k y (x, y dxdy Are these orthonormal? Example
3 Example Are these orthonormal? /2 /2 /2 /2 /2 /2 -/2 -/2 /2 -/2 /2 -/2 /2 -/2 -/2 /2 Discrete Expansion Coefficients The discrete expansion is g[m,n] = l= c k,l k= b k,l [m,n] If the basis functions b k,l [m,n] are orthonormal then c k,l = b k,l,g = = m= n= = c k = l = m= n= b k,l [m,n]g[m,n] k, l b k, k = l = b k,l [m,n] c k, l b k,l [m,n] b k, m= n= = c k, l δ[k k,l l ] k = = c k,l l = l [m,n] l [m,n] Continuous Expansion Coefficients The continuous expansion is g(x, y = ( c k x b kx (x, ydk x dk y If the basis functions b kx (x, y are orthonormal then ( = b kx,g = b kx c k x (x, yg(x, ydxdy ( = b kx (x, y c k x, k b y k x, k y (x, yd k x d k y dxdy ( k y = c k x, ( ( b kx (x, yb k x, k y (x, ydxdyd k x d k y = c k x, k δ(k y x k x k y d k x d k y = c k x 3
4 Separable Basis Functions Discrete b k,l [m,n] = b k [m]b l [n] e.g. δ[m k,n l] = δ[m k]δ[n l] Continuous b kx (x, y = b kx (xb ky (y e.g. δ( x x i,y y i = δ( x x i δ y y i ( Separable Basis Functions b [n] = [ 0] b 2 [n] = [0 ] b [m] = 0 b 0 0 [m] = 0 b [n] = [ 0] b 2 [n] = [0 ] b 2 [m] = 0 b 2 [m] = Separable Basis Functions b [n] = [ ]/ 2 b 2 [n] = [ ]/ 2 b [m] = /2 /2 /2 -/2 2 b [m] = /2 /2 2 /2 -/2 b [m] = 2 b [n] = [ ]/ 2 /2 /2 -/2 -/2 b 2 [n] = [ ]/ 2 /2 -/2 b [m] = 2 -/2 /2 b k,l [m,n] = b k [m] b l [n] b k [m] = exp( πmk / 2 b l [n] = exp( πnl / 2 4
5 Example x = Basis Functions Coefficients Sum Object Fourier Basis Functions Recall that for D the basis functions are complex exponentials b kx ( x = e j 2πk xx For 2D, we use the separable 2D functions b kx ( x, y = b kx ( xb ky y ( = e j 2πk xx e j 2πk yy = e j 2π (k xx +k y y Are they orthonormal? Plane Waves e j 2π (k xx +k y y = cos( 2π(k x x + k y y + j sin( 2π(k x x + k y y k x 2 + k y 2 /k y /k x cos(2πk x x cos(2πk y y cos(2πk x x +2πk y y 5
6 Plane Waves A θ /k y D θ B /k x C ΔABC ~ ΔBDC AC BC = AB BD BD = AB BC AC = θ = arctan k y k x k x k y k x 2 + k y 2 = k x 2 + k y 2 Fourier Transform 2D Fourier Transform = F[ g( x,y ] = e j 2π k x x +k y y (,g = g(x,y ( dxdy e j 2π k x x +k y y Inverse Fourier Transform g(x, y = ( dk x dk y e j 2π k x x +k y y Separable Functions g( x, y is said to be a separable function if it can be written as g( x, y = g X ( xg Y ( y The Fourier Transform is then separable as well. = g(x, y e j 2π ( k x x +k y y dxdy = g X ( xe j 2πk x x dx g Y ( y e j 2πkyy dy = G X (k x G Y (k y Example g(x, y = Π(xΠ(y = sinc(k x sinc(k y 6
7 Example g(x, y = Π(xΠ(y = sinc(k x sinc(k y Example (sinc/rect y /2 -/2 /2 x -/2 Example (sinc/rect 7
8 8
9 g(x, y = δ(x, y = δ(xδ(y = g(x, y = δ(x = δ(k y!!! g(x, y =+ e j 2πax = δ k x ( + δ(k x + aδ(k y g(x, y =+ e j 2πay = δ(k x + δ(k x δ(k y a g(x, y = cos(2π(ax + by = 2 δ(k x aδ(k y b + 2 δ(k x + aδ(k y + b 9
10 = δ(k x + g(x, y =??? δ(k x + cδ(k y + δ(k x δ(k y d + 2 δ(k x aδ(k y b + 2 δ(k x + aδ(k y + b 0
11 Basic Properties Linearity F[ ag(x, y + bh(x, y ] = a + bh(k x Scaling F[ g(ax,by ] = ab G k x a, k x b Shift F[ g(x a, y b ] = e j 2π (k xa +k yb Modulation j 2π (xa +yb [ ] = a b F g(x, ye Modulation Example x = * = Convolution/Multiplication Convolution F[ g(x, y h(x,y ] = H(k x Multiplication F[ g(x, yh(x,y ] = H(k x Multiplication in one domain translates into convolution in the other domain.
12 Convolution Example F[ g (x, y ] = G ( k x, k y Symmetry If g(x, y is real then g(x, y = g (x, y, so G ( k x, k y = k y k x Parseval s Relations Energy is preserved 2 2 g(x, y dxdy = G(kx dkx dk y So is the inner product g,h = G,H g (x,y h(x,ydxdy = G (k x H(k x dk x dk y 2
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